Chapter 3: Linear regression

date()

## [1] "Mon Nov 27 12:22:31 2023"

Data

Let’s import data. We will use “Boston” data from MASS package.

library(MASS)

## Warning: package 'MASS' was built under R version 4.2.3

library(ggplot2)

## Warning: package 'ggplot2' was built under R version 4.2.3

data("Boston")

Now we can explore how the data looks

str(Boston)

## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...

dim(Boston)

## [1] 506  14

Dataset “Boston” has 506 observations and 14 variables. Most of the varables are numeric, but two are integrals. Variables of the data and their diecription are presented bellow.

## Warning: package 'knitr' was built under R version 4.2.3

Variable	Descripition
crim	per capita crime rate by town.
zn	proportion of residential land zoned for lots over 25,000 sq.ft
indus	proportion of non-retail business acres per town.
chas	Charles River dummy variable
nox	nitrogen oxides concentration
rm	average number of rooms per dwelling.
age	proportion of owner-occupied units built prior to 1940.
dis	weighted mean of distances to five Boston employment centres.
rad	index of accessibility to radial highways.
tax	full-value property-tax rate per $10,000.
ptratio	pupil-teacher ratio by town.
black	proportion of blacks by town
lstat	lower status of the population
medv	median value of owner-occupied homes in $1000s.

Summary of the data

##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08205   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

Here you can see the pair-wise comparisation of each variable. Matrix is fairly small in size but provides some visual insigts.

One can see that some variables have little relationships (such as age and ptratio) whereas some have strong relationship (nox and dis). Correlation matrix could be easier to interpret.

## Warning: package 'corrplot' was built under R version 4.2.3

## corrplot 0.92 loaded

##Scaling the data

boston_scaled= scale(Boston)
boston_scaled=as.data.frame(boston_scaled)
summary(boston_scaled)

##       crim                 zn               indus              chas        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109   Median :-0.2723  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648  
##       nox                rm               age               dis         
##  Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658  
##  1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049  
##  Median :-0.1441   Median :-0.1084   Median : 0.3171   Median :-0.2790  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617  
##  Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566  
##       rad               tax             ptratio            black        
##  Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033  
##  1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049  
##  Median :-0.5225   Median :-0.4642   Median : 0.2746   Median : 0.3808  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332  
##  Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406  
##      lstat              medv        
##  Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 3.5453   Max.   : 2.9865

After scaling the mean of each variable is 0. So instead of absolute values the value suggests how far from the mean within that variable the observation is.

bins <- quantile(boston_scaled$crim)
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)

divide data

n <- nrow(boston_scaled)
ind <- sample(n,  size = n * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]

Linear discriminant analysis

lda.fit <- lda(crime ~ ., data = train)

lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  graphics::arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(train$crime)

# plot the lda results (select both lines and execute them at the same time!)
plot(lda.fit, dimen = 2, col=classes, pch=classes)
lda.arrows(lda.fit, myscale = 2)

Here we can find see what variables separates the data into different groups. It seems that rad has the greatest effect on LD1 where as nox has the greatest effect on LD2. If i interpret the graph correctly, access to high ways is linked to high crime rate and nitrogen oxides to medium high rates.

correct_classes <- test$crime
test <- dplyr::select(test, -crime)
lda.pred <- predict(lda.fit, newdata = test)
table(correct = correct_classes, predicted = lda.pred$class)

##           predicted
## correct    low med_low med_high high
##   low       14      13        3    0
##   med_low    4      18        6    0
##   med_high   0       5       15    0
##   high       0       0        0   24

In the most cases the model made correct predictions. Especially success rate on predicting high crime rate was good. Predicting medium low crimes seems to be the most difficult. How ever the predictions are not perfect: for example in one case the model predicted medium high crime rate even tough actual rate was low.

data(Boston)
scal_bos=as.data.frame(scale(Boston))
dist_eu <- dist(scal_bos)
summary(dist_eu)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

##K-means

Let’s calculate k-means

km <- kmeans(scal_bos, centers = 5)
summary(km)

##              Length Class  Mode   
## cluster      506    -none- numeric
## centers       70    -none- numeric
## totss          1    -none- numeric
## withinss       5    -none- numeric
## tot.withinss   1    -none- numeric
## betweenss      1    -none- numeric
## size           5    -none- numeric
## iter           1    -none- numeric
## ifault         1    -none- numeric

How ever the means calculated above would be better if we would fing out the optimal number of clusetrs. So let us do that!

k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(scal_bos, k)$tot.withinss})
qplot(x = 1:k_max, y = twcss, geom = 'line')

## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

km <- kmeans(scal_bos, centers = 2)

pairs(scal_bos, col = km$cluster)

It is hard to see what is going on, so let’s take a closer look on some of interesting looking variables

pairs(scal_bos[c(1,10)], col = km$cluster)

pairs(scal_bos[c(1,7)], col = km$cluster)

For example on these two plots one can see clear clustering with crime and tax. High tax and high crime rate seems to create two different clusters. This suggests that there are less crime in areas with high tax.

It also seems that crime rate increases in with owner-occupied units built prior to 1940.

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Bonus!

km <- kmeans(scal_bos, centers = 4)
lda.fit <- lda(km$cluster ~ ., data = scal_bos)

lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  graphics::arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# plot the lda results (select both lines and execute them at the same time!)
plot(lda.fit, dimen = 2, col=classes, pch=classes)
lda.arrows(lda.fit, myscale = 2.5)

The most influential variable is black on LD2. On LD1 it is hard to determine, but it might be nox.

super bonus!

library(plotly)

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:MASS':
## 
##     select

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

model_predictors <- dplyr::select(train, -crime)
lda.fit <- lda(crime ~ ., data = train)
# check the dimensions
dim(model_predictors)

## [1] 404  13

dim(lda.fit$scaling)

## [1] 13  3

# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)


plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color= train$crime)

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